This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here's how to solve these polynomial problems. Given the polynomial P(x) = 2x^3 - 5x^2 - 19x + 42. a) Find the quotient and the remainder for the division of P(x) by 2x-7. We will use polynomial long division. Step 1: Set up the long division. r x^2 + x - 6 \\[-3pt] 2x-7 ) 2x^3 - 5x^2 - 19x + 42 \\[-3pt] Step 2: Divide the leading terms. Divide 2x^3 by 2x to get x^2. Multiply x^2 by (2x-7) and subtract. r x^2 +x-6 \\[-3pt] 2x-7 ) 2x^3 - 5x^2 - 19x + 42 \\[-3pt] -(2x^3 - 7x^2) \\[-3pt] 2x^2 - 19x \\[-3pt] Step 3: Repeat the process. Bring down the next term (-19x). Divide 2x^2 by 2x to get x. Multiply x by (2x-7) and subtract. r x^2 + x -6 \\[-3pt] 2x-7 ) 2x^3 - 5x^2 - 19x + 42 \\[-3pt] -(2x^3 - 7x^2) \\[-3pt] 2x^2 - 19x \\[-3pt] -(2x^2 - 7x) \\[-3pt] -12x + 42 \\[-3pt] Step 4: Repeat again. Bring down the next term (+42). Divide -12x by 2x to get -6. Multiply -6 by (2x-7) and subtract. r x^2 + x - 6 \\[-3pt] 2x-7 ) 2x^3 - 5x^2 - 19x + 42 \\[-3pt] -(2x^3 - 7x^2) \\[-3pt] 2x^2 - 19x \\[-3pt] -(2x^2 - 7x) \\[-3pt] -12x + 42 \\[-3pt] -(-12x + 42) \\[-3pt] 0 \\[-3pt] The quotient is x^2 + x - 6 and the remainder is 0. The quotient is x^2 + x - 6 and the remainder is 0. b) Factorise P(x) completely. Since the remainder from part (a) is 0, (2x-7) is a factor of P(x). So, P(x) = (2x-7)(x^2 + x - 6). Step 1: Factor the quadratic term. We need to factor the quadratic expression x^2 + x - 6. We look for two numbers that multiply to -6 and add to 1. These numbers are 3 and -2. So, x^2 + x - 6 = (x+3)(x-2). Step 2: Write the completely factored form. Substitute the factored quadratic back into the expression for P(x). P(x) = (2x-7)(x+3)(x-2) The completely factored form is (2x-7)(x+3)(x-2). c) Solve if P(x) = 0. To solve P(x) = 0, we set the factored form of P(x) equal to zero. (2x-7)(x+3)(x-2) = 0 Step 1: Set each factor to zero and solve for x. For the first factor: 2x-7 = 0 2x = 7 x = (7)/(2) For the second factor: x+3 = 0 x = -3 For the third factor: x-2 = 0 x = 2 The solutions are x = (7)/(2), x = -3, x = 2. d) Solve: 2x^3 + 9x^2 - 2x - 24 = 0. Let Q(x) = 2x^3 + 9x^2 - 2x - 24. We need to find the roots of this cubic equation. Step 1: Find a rational root using the Rational Root Theorem or by testing values. We test integer factors of the constant term (-24) divided by integer factors of the leading coefficient (2). Let's try x = -2: Q(-2) = 2(-2)^3 + 9(-2)^2 - 2(-2) - 24 Q(-2) = 2(-8) + 9(4) + 4 - 24 Q(-2) = -16 + 36 + 4 - 24 Q(-2) = 40 - 40 Q(-2) = 0 Since Q(-2) = 0, x = -2 is a root, and (x+2) is a factor of Q(x). Step 2: Use synthetic division to divide Q(x) by (x+2). The coefficients of Q(x) are 2, 9, -2, -24. c|cccc -2 & 2 & 9 & -2 & -24 \\ & & -4 & -10 & 24 \\ & 2 & 5 & -12 & 0 \\ The quotient is 2x^2 + 5x - 12. So, Q(x) = (x+2)(2x^2 + 5x - 12) = 0. Step 3: Factor the quadratic quotient. We need to solve 2x^2 + 5x - 12 = 0. We can factor this quadratic. We look for two numbers that multiply to 2 × -12 = -24 and add to 5. These numbers are 8 and -3. 2x^2 + 8x - 3x - 12 = 0 2x(x+4) - 3(x+4) = 0 (2x-3)(x+4) = 0 Step 4: Set each factor to zero and solve for x. From the first factor we found: x+2 = 0 x = -2 From the quadratic factors: 2x-3 = 0 2x = 3 x = (3)/(2) x+4 = 0 x = -4 The solutions are x = -2, x = (3)/(2), x = -4. 3 done, 2 left today. You're making progress.